The Michigan Mathematical Journal

On Diamond’s L1 Criterion for Asymptotic Density of Beurling Generalized Integers

Gregory Debruyne and Jasson Vindas

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We give a short proof of the L1 criterion for Beurling generalized integers to have a positive asymptotic density. We in fact prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate m(x)=nkxμ(nk)/nk=o(1) with the Beurling analog μ of the Möbius function.

Article information

Source
Michigan Math. J., Volume 68, Issue 1 (2019), 211-223.

Dates
Received: 12 April 2017
Revised: 3 November 2017
First available in Project Euclid: 31 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1548903624

Digital Object Identifier
doi:10.1307/mmj/1548903624

Mathematical Reviews number (MathSciNet)
MR3934610

Zentralblatt MATH identifier
07155464

Subjects
Primary: 11N80: Generalized primes and integers
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11M45: Tauberian theorems [See also 40E05] 11N37: Asymptotic results on arithmetic functions

Citation

Debruyne, Gregory; Vindas, Jasson. On Diamond’s $L^{1}$ Criterion for Asymptotic Density of Beurling Generalized Integers. Michigan Math. J. 68 (2019), no. 1, 211--223. doi:10.1307/mmj/1548903624. https://projecteuclid.org/euclid.mmj/1548903624


Export citation

References

  • [1] A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, Acta Math. 68 (1937), 255–291.
  • [2] H. Bremermann, Distributions, complex variables and Fourier transforms, Addison-Wesley, Reading, Massachusetts, 1965.
  • [3] G. Debruyne, H. G. Diamond, and J. Vindas, $M(x)=o(x)$ Estimates for Beurling numbers, J. Théor. Nombres Bordeaux 30 (2018), 469–483.
  • [4] G. Debruyne and J. Vindas, Generalization of the Wiener–Ikehara theorem, Illinois J. Math. 60 (2016), 613–624.
  • [5] G. Debruyne and J. Vindas, On PNT equivalences for Beurling numbers, Monatsh. Math. 184 (2017), 401–424.
  • [6] G. Debruyne and J. Vindas, Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior, J. Anal. Math. (to appear), arXiv:1604.05069.
  • [7] H. G. Diamond, When do Beurling generalized integers have a density? J. Reine Angew. Math. 295 (1977), 22–39.
  • [8] H. G. Diamond and W.-B. Zhang, Beurling generalized numbers, Math. Surv. Monogr., 213, American Mathematical Society, Providence, RI, 2016.
  • [9] J.-P. Kahane, Sur les nombres premiers généralisés de Beurling. Preuve d’une conjecture de Bateman et Diamond, J. Théor. Nombres Bordeaux 9 (1997), 251–266.
  • [10] J.-P. Kahane, Le rôle des algèbres $A$ de Wiener, $A^{\infty }$ de Beurling et $H^{1}$ de Sobolev dans la théorie des nombres premiers généralisés de Beurling, Ann. Inst. Fourier (Grenoble) 48 (1998), 611–648.
  • [11] J.-P. Kahane, Conditions pour que les entiers de Beurling aient une densité, J. Théor. Nombres Bordeaux 29 (2017), 681–692.
  • [12] J.-P. Kahane and É. Saïas, Sur l’exemple d’Euler d’une fonction complètement multiplicative à somme nulle, C. R. Math. Acad. Sci. Paris 354 (2016), 559–561.
  • [13] J.-P. Kahane and É. Saïas, Fonctions complètement multiplicatives de somme nulle, Expo. Math. 35 (2017), 364–389.
  • [14] J. Korevaar, Distributional Wiener–Ikehara theorem and twin primes, Indag. Math. (N.S.) 16 (2005), 37–49.
  • [15] W. Rudin, Lectures on the edge-of-the-wedge theorem, CBMS Reg. Conf. Ser. Math., 6, AMS, Providence, RI, 1971.
  • [16] J.-C. Schlage-Puchta and J. Vindas, The prime number theorem for Beurling’s generalized numbers. New cases, Acta Arith. 153 (2012), 299–324.
  • [17] W.-B. Zhang, Extensions of Beurling’s prime number theorem, Int. J. Number Theory 11 (2015), 1589–1616.